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Re: exponential cooling - a short exercise



To: phys-l
1. If you have "Physics, " vol. 1, fourth edition, by Resnick, Halliday, and Krane, you
might want to look at problem 9 in chapter 22.

2. Also, my summary statement: in general, the temperataure of an
object does not decay exponentially with time. However, the decay is
approximately exponential if the object is not too hot or cold
compared to its surroundings. The object must not be so large or its
conductivity so small that temperature variations within the object
become important.

Jack Uretsky wrote:

I think that intuituion and the art of estimation walk hand-in-hand.
The exponential cooling problem provides a nice example, in my opinion.
So try this:
The cooling of a radiating (black) body of temperature T,
immersed in a radiation field of black-body temperature T_0, is
given by the equation:
dT/dt = -k[T^4 -(T_0)^4]
where k is some constant.
1. Show that if T is close to T_0, then the cooling is
exponential in time with inverse time constant 4k(T_0)^3, and
calculate the correction to this expression proportional to (T-T_0).
2. Show that if T is much larger than T_0, then the
temperature decrease is inversely proportional to the time and the
proportionality constant is -k(T_i)^4. Correction: I mean the
decrease is proportional to the time (sorry, but this is a lousy editor).






Steven T. Ratliff
Associate Professor of Physics
Northwestern College
3003 Snelling Av. N.
Saint Paul, MN 55113-1598

Internet: stratliff@nwc.edu (or str@nwc.edu)